Featured Speaker :  Jacopo Borga (Zurich)

Title :  Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Abstract :  Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called tandem walks, are well-known to be related to each other through several bijections. In order to study their scaling and local limits, we introduce a further new family of discrete objects, called coalescent-walk processes and we relate them with the other previously mentioned families introducing some new bijections.

We prove joint Benjamini-Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure of the unit square, called the Baxter permuton, and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. We further relate the limiting objects of the four families to each other, both in the local and scaling limit case.

To prove the scaling limit result, we show that the associated random coalescent-walk process converges in distribution to the coalescing flow of a perturbed version of the Tanaka stochastic differential equation. This result has connections with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations.

This is a joint work with Mickael Maazoun.